Question

The true set of real values of $x$ for which the function, $f\left(x\right)=x\ln x(x+1)$ is positive is:

• A. $(0,\infty )$
• B. $(\frac{1}{e},\infty )$
• C. $[e,\infty )$
• D. $(0,1)\cup (1,\infty )$

Answer 1

The correct option is A $(0,\infty )$

$f\left(x\right)=x\ln x(x+1)$

$f\left(x\right)=>0$

means $x>0$

Hence

$ln(x+1)x>0$

which is only positive when

$x(x+1)>0$

hence $xϵ(-\infty ,-1)\cup (0,\infty )$

but $x>0$

Hence $xϵ(0,\infty )$.